OK here is what I think I know! (leaving room to be wrong!)
Yes, it has to be the same time but you can pick any time. So, if you get up around 6, you always temp (even on you days off!) at about 7am. Personally, I sleep with my mouth open (not too pretty for my DH!) so I wake up, realize my mouth is open, then close it and put my pillow under my chin to keep it shut and go back to sleep for a bit while it "warms up", so to speak. What you are trying to do is see a pattern so the goal is to keep everything as 'the same' as possible.
So, it's no problem if you go in and out of sleep: just set your alarm for an early time, take it and go back to sleep.
When I started I had to write it down in the morning but now I always remember so I can just go back to sleep.
Also: Get a digital thermometer and one that remembers your last temp so that you can re-check if you do forget.
Once you start doing it, if you are like me, you'll get kind of addicted! It's cool! You can see when you ovulate, how things are going, a few days before you get af... I really like it. I have seen that it was smart to BD before the DR even tells me. (I get a big dip when I ovulate)
It does take a few months to see the patterns and it does not work for everyone and it not for everyone, just so you know. I also really recommend "Fertility Friend" online. It's free: they want you to sign up for a paid one but you don't really need it, they'll give you a free trail so see what you thing.
OK, hope this helps, happy to give you any more info if you need it!